The CFPB, which was created by the 2010 Dodd-Frank financial overhaul, is not funded through the annual appropriations process. Instead, it receives money through transfers from the Federal Reserve.

Moore said it was necessary to keep Congress out of funding for the CFPB so that it would not be subject to politics.

"It is well documented that Congress wanted its funding to be free of political influence when it created the Bureau. In order to protect the consumers, it needed to be free of political influence," Moore said.

But House Republicans have long argued that subjecting the CFPB to the annual appropriations process would increase congressional oversight of the agency.

"Congressional oversight makes agencies both more responsive and more responsible," said Rep. Ander Crenshaw (R-Fla.), the chairman of the House Appropriations Financial Services subcommittee.

Rep. Jose Serrano (D-N.Y.), the top Democrat on the House Appropriations Financial Services subcommittee, broke from his colleagues. Serrano warned that the lack of congressional oversight of Wall Street and its enforcement agencies ultimately led to the financial crisis.

"It sounds great for many members to have an agency be on its own and do the right thing. But past history shows us that when we did that, when we did not supervise, and when we did not have oversight, it did just the opposite," Serrano said.